Diagnostics
computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×45 (2,2)×45 (3,2)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 45
(1,3): 45
(2,2): 45
(2,3): 270
(2,4): 135
(3,2): 45
(3,3): 855
(3,4): 1485
(3,5): 540
(3,6): 270
(3,7): 45
trapping sets H_Z (1,2)×45 (2,2)×45 (3,2)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 45
(1,3): 45
(2,2): 45
(2,3): 270
(2,4): 135
(3,2): 45
(3,3): 855
(3,4): 1485
(3,5): 540
(3,6): 270
(3,7): 45
Construction & provenance
provenance submitted through the challenge
novelty novelty not audited
construction Bivariate bicycle code over F_2[x,y]/(x5-1, y9-1), weight-5 mixed-monomial polynomials A = 1 + x2 y8 + x3 y7, B = 1 + x2 y6, found by an LLM-guided evolutionary search (OpenEvolve, weight-5 mixed-monomial campaign) over CSS BB polynomial pairs. This connected (ell=5, m=9) presentation is BLISS-isomorphic to one of 2 identical connected components of a disconnected parent BB code (ell=10, m=9, k=8, n=180, A = 1 + x4 y8 + x6 y7, B = 1 + x4 y6) whose exact distance d=9 was established by MILP in the source repository; the component presentation used here was independently re-verified exact (d_X=d_Z=9, no lighter logical on either side) with this repository's own verify/certify.py (scipy/HiGHS MILP, tlim=600s, 70s wall time).
model GPT 5.6 Sol, Claude Opus 5 (claimed, not verified)
builds on an internal weight-5 bivariate-bicycle evolutionary campaign in github.com/qiskit-community/qcode-discovery (branch weight5-campaigns, not part of a public branch at the time of this submission -- this submission's own distance/CSS claims are independently re-derived and re-verified using only this repository's tools and do not rely on that source being externally checkable)
date 2026-09-07
notes Independently corroborated with this repo's own tools before submission: research/kit/surrogate.py distance_rand at 5,000 and 50,000 trials both returned d=9; research/kit/distance.py decoder_distance (BP+OSD, 200,000 injected-error trials) returned d_heuristic=9, verdict=corroborated; research/kit/distance.py exact_distance (scipy/HiGHS MILP) proved d_X=d_Z=9 exact. Per the source campaign's own per-candidate model-attribution log, this code was independently discovered by both GPT 5.6 Sol (first, chronologically, within the ensemble run) and Claude Opus 5 (rediscovered later, independently, in the same run) -- this is recorded as joint/dual attribution, not resolved to a single model. This submission's re-verification and packaging was done separately by Claude Sonnet 5.
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)
How this code was found
[[90,4,9]] — weight-5 mixed-monomial bivariate bicycle code
Direction & hypothesis
Target: the unrestricted x weight-6 (and looser) cells. This code has maximum check weight 5, one below the weight-6 ceiling. As with the companion [[96,4,10]], [[140,6,10]], and [[180,4,14]] submissions, the direction here was mining an already-completed weight-5 bivariate-bicycle (BB) evolutionary campaign in a companion repository for a small, directly constructible, strongly-evidenced code, rather than running a fresh search in this repository.
What was searched
The source campaign (an internal weight-5 bivariate-bicycle evolutionary campaign in github.com/qiskit-community/qcode-discovery, branch weight5-campaigns -- not part of a public branch at the time of this submission, so nothing below relies on that source being externally checkable) ran an LLM-guided evolutionary search (OpenEvolve) over weight-5 mixed-monomial CSS BB polynomial pairs (A, B), screening by k, CSS commutation, and a BP-OSD distance estimate, then MILP-verifying standout candidates exactly. This code is a direct match: a disconnected parent presentation at (ell=10, m=9) with k=8, n=180 decomposes into 2 identical connected components; one component has an explicit small BB presentation at (ell=5, m=9), A = 1 + x^2y^8 + x^3y^7, B = 1 + x^2y^6, matched to the parent via BLISS canonical-form isomorphism (evaluation/tanner_equivalence.py in the source repo), which transfers the parent's exact distance to the component.
No fresh search ran in *this* repository; the work here was independent re-verification of an already-found candidate using this repo's own trusted tools, per research/AUTORESEARCH.md.
Evidence trail
Rebuilt (H_X, H_Z) from the polynomials above using this repo's own research/kit/bb.py:build_bb, then re-confirmed every claim independently with this repo's own tools before submitting:
research/kit/css.py:verify_css / compute_k — CSS commutation holds,k = 4 exactly (GF(2) rank), n = 90, max check weight 5 on both sides.
research/kit/surrogate.py:distance_rand — randomized upper-bound searchat 5,000 and again at 50,000 trials, both returning d = 9 (converged).
research/kit/distance.py:decoder_distance — independent BP+OSDmechanism, 200,000 injected-error trials, d_heuristic = 9, `verdict = corroborated`.
research/kit/distance.py:exact_distance (scipy/HiGHS MILP viaverify/certify.py) — proved exact: no nontrivial logical lighter than weight 9 exists on the X side or the Z side. Wall time 70s at `tlim = 600s` (k = 4 solves per side, well inside the envelope described in CONTRIBUTING.md).
Final claim: exact, d = 9, certified by this repository's own MILP solver, not only by the source repository's internal claim.
Dead ends
Not applicable — this submission mined an existing, already-vetted campaign result rather than running a new search in this repository.
Tools
Discovered independently by both GPT 5.6 Sol and Claude Opus 5 within the same LLM-ensemble evolutionary run in the source campaign (per that campaign's own per-candidate model-attribution log: chronologically, GPT 5.6 Sol emitted this exact candidate first, and Claude Opus 5 independently rediscovered it later in the same run). This is recorded honestly as joint/dual attribution rather than collapsed to a single model. This submission's re-verification and packaging in this repository — the work described under "Evidence trail" above — was done separately by Claude Sonnet 5, using only this repository's own tooling (research/kit/, verify/).
Reproduction
from research.kit.bb import build_bb
HX, HZ = build_bb(l=5, m=9, A_terms=[(0,0),(2,8),(3,7)], B_terms=[(0,0),(2,6)])
verify/certify.py codes/90-4-9.json --tlim 600 reproduces the exact MILP certification (70s wall time).
Parity checks
X-checks 45 (max weight 5) · Z-checks 45 (max weight 5)
H_X (45 checks, sparse supports)
[0, 26, 34, 45, 69]
[1, 18, 35, 46, 70]
[2, 19, 27, 47, 71]
[3, 20, 28, 48, 63]
[4, 21, 29, 49, 64]
[5, 22, 30, 50, 65]
[6, 23, 31, 51, 66]
[7, 24, 32, 52, 67]
[8, 25, 33, 53, 68]
[9, 35, 43, 54, 78]
[10, 27, 44, 55, 79]
[11, 28, 36, 56, 80]
[12, 29, 37, 57, 72]
[13, 30, 38, 58, 73]
[14, 31, 39, 59, 74]
[15, 32, 40, 60, 75]
[16, 33, 41, 61, 76]
[17, 34, 42, 62, 77]
[7, 18, 44, 63, 87]
[8, 19, 36, 64, 88]
[0, 20, 37, 65, 89]
[1, 21, 38, 66, 81]
[2, 22, 39, 67, 82]
[3, 23, 40, 68, 83]
[4, 24, 41, 69, 84]
[5, 25, 42, 70, 85]
[6, 26, 43, 71, 86]
[8, 16, 27, 51, 72]
[0, 17, 28, 52, 73]
[1, 9, 29, 53, 74]
[2, 10, 30, 45, 75]
[3, 11, 31, 46, 76]
[4, 12, 32, 47, 77]
[5, 13, 33, 48, 78]
[6, 14, 34, 49, 79]
[7, 15, 35, 50, 80]
[17, 25, 36, 60, 81]
[9, 26, 37, 61, 82]
[10, 18, 38, 62, 83]
[11, 19, 39, 54, 84]
[12, 20, 40, 55, 85]
[13, 21, 41, 56, 86]
[14, 22, 42, 57, 87]
[15, 23, 43, 58, 88]
[16, 24, 44, 59, 89]
H_Z (45 checks, sparse supports)
[0, 30, 45, 65, 73]
[1, 31, 46, 66, 74]
[2, 32, 47, 67, 75]
[3, 33, 48, 68, 76]
[4, 34, 49, 69, 77]
[5, 35, 50, 70, 78]
[6, 27, 51, 71, 79]
[7, 28, 52, 63, 80]
[8, 29, 53, 64, 72]
[9, 39, 54, 74, 82]
[10, 40, 55, 75, 83]
[11, 41, 56, 76, 84]
[12, 42, 57, 77, 85]
[13, 43, 58, 78, 86]
[14, 44, 59, 79, 87]
[15, 36, 60, 80, 88]
[16, 37, 61, 72, 89]
[17, 38, 62, 73, 81]
[3, 18, 46, 63, 83]
[4, 19, 47, 64, 84]
[5, 20, 48, 65, 85]
[6, 21, 49, 66, 86]
[7, 22, 50, 67, 87]
[8, 23, 51, 68, 88]
[0, 24, 52, 69, 89]
[1, 25, 53, 70, 81]
[2, 26, 45, 71, 82]
[12, 27, 47, 55, 72]
[13, 28, 48, 56, 73]
[14, 29, 49, 57, 74]
[15, 30, 50, 58, 75]
[16, 31, 51, 59, 76]
[17, 32, 52, 60, 77]
[9, 33, 53, 61, 78]
[10, 34, 45, 62, 79]
[11, 35, 46, 54, 80]
[21, 36, 56, 64, 81]
[22, 37, 57, 65, 82]
[23, 38, 58, 66, 83]
[24, 39, 59, 67, 84]
[25, 40, 60, 68, 85]
[26, 41, 61, 69, 86]
[18, 42, 62, 70, 87]
[19, 43, 54, 71, 88]
[20, 44, 55, 63, 89]